Friday, March 21, 2014

Range Equity vs Range Balance -- Which matters more?

Often people use the equity of their hand or their range vs. their opponents range as a way to gauge how good or bad a certain situation is for them. While focusing on equity is somewhat intuitive and convenient (because it is easy to calculate), it completely ignores the impact of your strategic options, or lack thereof, on your odds of winning the hand.

In turns out that even against extremely simple strategies, there are ranges with good equity that cannot possibly win their share of the pot. Equity assumes that your opponent lets you see all 5 board cards and that they let you show down your hand.In practice, even weak players will put way too much pressure on you to make this assumption realistic.

Often people think that when they put themselves in situations with capped or unbalanced ranges that the situation is just "tough to play", but that if they were better and able to play perfectly they would be able to defend their hand equity. As we'll see in the examples below, this simply isn't true. Unbalanced and particularly capped ranges are just fundamentally unable to defend their equity, even against opponents who play very simple, predictable strategies.

Range balance is a much more important factor than range equity in determining how much of a pot you are likely to be able to win on average on the river. The deeper you are, and the better your opponent plays, the more balance matters and the less equity matters. As stacks get shorter, equity becomes the dominating factor, so for SnG / MTT players it is more reasonable to rely on equity in your analysis.

The simplest example, the nuts/air vs. made hand

I'm going to start by looking at a contrived example that illustrates the basic ideas that prevent unbalanced ranges from defending their equity. This example is obviously much simpler than any real poker scenario, but it captures the strategic essence of many situations.

We're heads up on the river and player 1 has played in such a way that his hand is face up as a medium strength made hand. Player 2 has taken a very aggressive line that restricts his range only to nuts or air. Let's assume player 1 is in position.

We'll assume that, on the river, player 2 has the nuts 50% and air 50%. This means that both players have exactly 50% equity on the river. Furthermore, assume the pot is 60bb and there is 120bb left behind to bet. What is each player's EV?

If we just focused on equity, obviously each player would expect to win half of the 60bb pot on average for an EV of 30bb. However, even an extremely simple and brain dead strategy for player 2 does much better than this.

Suppose player 2 plays an extremely simple (and quite fishy) strategy and just over-bet shoves with the nuts 100% of the time and with air 50% of the time and check folds his air the other 50%. Player 1 must fold to the shove, as he's losing 2/3rds of the time he calls and 2/3 * -120bb + 1/3 * 180bb = -20bb so calling is clearly -EV. This simple, fishy strategy lets player 2 win the pot 75% of the time, even though his range only has 50% equity. Even if player 1 is a much better player, there is nothing he can do against this simple strategy.

When you actually are playing in this situation as player 1, it can feel like a complex leveling battle of, "he knows my hand is face up, so he's going to try and bluff me off it, but I know, that he knows, and he knows that I know that he knows" and you can sit there agonizing over what level your opponent is on. It's easy to convince yourself that if you could somehow win the leveling war then you could win the pot half the time on this river.

The reality of the situation is that because your range is unbalanced and your opponents range is not, he has a fundamental advantage over you, choice. He can take different actions with different portions of his range so that even though his average equity is 50%, he can use the strong portion of his range (the nuts) to protect some of the weak portion of his range (the air) and make them indistinguishable to you, thus increasing his EV.

The advantage here for player 2 is NOT due to his opponents hand being face up, rather it is due to the fact that he knows whether he actually has the nuts or air and his opponent does not.

To see this, take the exact same scenario as above, but force player 2 to act without looking at his cards. From player 1's perspective he still has no idea whether his opponent has the nuts or air. From player 2's perspective, he still knows exactly what his opponent has, player 1's hand is just as face up. However, in this case, it is easy for player 1 to win the pot half the time. When player 2 is playing blind, player 1 can just call any bet no matter the amount (or check) and he will win average winning exactly 50% of the pot, even though player 1's hand is still face up.

When you hold a balanced range (a range with a variety of hand strengths) in any given hand you know which specific portion of your range you actually hold and your opponent does not. That informational difference about your own holdings can be converted into money, often with very simple strategies as shown above, and your opponent can not prevent that, even with perfect play.

Obviously if a simple, over-bet shove 2x pot or fold strategy lets player 2 win 75% of the pot on average, no matter what his opponent does, the Nash Equilibrium strategy will do even better for him. As you can see in the video below, in a simple scenario where each player can bet 50% pot or shove all in at each point, player 2 (the hero) has an EV of 50bb in this scenario or about 84.4% of the pot, even though his equity is 50%. Furthermore, the deeper the stacks, the more player 2's EV increases. With 1000bb stacks player 2 wins 95% of the pot and as stacks go to infinity, player 2 wins 100% (if anyone is interested in a proof of that, ask in the comments).

Conclusions

Anytime you rely on equity to determine the EV of a decision you are assuming that your opponent is going to let you showdown your hand which, in practice, ignores many of the biggest parts of poker. How well balanced your range is determines how many strategic options you have available and increases the informational balance you have (by knowing your exact holdings) over your opponent (who is putting you on a hand range). In lots of post flop situations, the impact of your range balance will outweigh the actual equity strength of your range.

There is a lot more analysis that can be done here, for example, how much does this result change if, rather than a pure nuts or air scenario, we look at two players whose ranges overlap, but where one players in generally more polarized. Or what about scenarios where both players can have air but one players range is capped? Can adding air to a range actually increase its EV by opening up more strategic options, etc? For now, I'm going to leave those questions for a future blog post, but if there are any particular questions that you'd like me to analyze, just let me know :)